Problem: What is the inverse of the function $g(x)=-3(x+6)$ ? $g^{-1}(x)=$
Answer: Let's start by replacing $g(x)$ with $y$. $y=-3(x+6)$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=-3(x+6)$, so the inverse relationship is $x=-3(y+6)$. Solving this equation for $y$ will give us an expression for $g^{-1}(x)$. $\begin{aligned} x&=-3(y+6)\\\\ -\dfrac{1}{3}x&=y+6\\\\ -\dfrac{1}{3}x-6&=y\\\\\\ \end{aligned}$ The inverse of the function is $g^{-1}(x)=-\dfrac{1}{3}x-6$. [I saw someone solve this problem by originally solving for x. Were they wrong?]